**Problem 8** Find \( z_x \) if \( z = \frac{1}{2x^2ay} + \frac{3x^{\frac{5}{2}}abc}{y} \). This problem requires the computation of the partial derivative of \( z \) with respect to \( x \). The expression for \( z \) involves two terms: a rational function and a term with a power of \( x \). **Problem 5**: Find \(\frac{dz}{dx}\) and \(\frac{dz}{dy}\) if \(z = (x^2 + x - y)^7\). This problem involves finding the partial derivatives of the function \(z = (x^2 + x - y)^7\) with respect to \(x\) and \(y\). ### Explanation: - **Partial Derivative \(\frac{dz}{dx}\)**: This involves differentiating the function with respect to \(x\), while keeping \(y\) constant. - **Partial Derivative \(\frac{dz}{dy}\)**: This involves differentiating the function with respect to \(y\), while keeping \(x\) constant. Understanding how to find these derivatives is crucial for applications in multivariable calculus, particularly when dealing with functions that depend on more than one variable.

Problem 8 Find \( z_x \) if \( z = \frac{1}{2x^2ay} + \frac{3x^{\frac{5}{2}}abc}{y} \). This problem requires the computation of the partial derivative of \( z \) with respect to \( x \). The expression for \( z \) involves two terms: a rational function and a term with a power of \( x \). Problem 5: Find \(\frac{dz}{dx}\) and \(\frac{dz}{dy}\) if \(z = (x^2 + x - y)^7\). This problem involves finding the partial derivatives of the function \(z = (x^2 + x - y)^7\) with respect to \(x\) and \(y\). ### Explanation: - Partial Derivative \(\frac{dz}{dx}\): This involves differentiating the function with respect to \(x\), while keeping \(y\) constant. - Partial Derivative \(\frac{dz}{dy}\): This involves differentiating the function with respect to \(y\), while keeping \(x\) constant. Understanding how to find these derivatives is crucial for applications in multivariable calculus, particularly when dealing with functions that depend on more than one variable.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Related questions

Q: 2. As an application of partial derivative of multi-variable functions, provide one page abou…

Q: What are Partial Derivatives?

Q: Given the function f(x,y) = y-6x. Find the domain & its boundary and determine if it is open,…

A: The function given is fx,y=y-6x. The objective is to determine The domain and its boundary. If it…

Q: Find the local maximum, minimum and saddle points of the function using the second derivative test:…

Q: Use partial derivatives to obtain the formula for the best least-squares fit to the data points.…

A: Solution

Q: Suppose y is a function of z. Choose the correct answer. Select one: dy a. 4 (z°y²) = x5 . 2y + y² -…

A: Derivative of y is dy/dx.

Q: Sketch the level curves of the function z(x, y) = 10- 9 8 7 6 5 4 3 -10 -9 -8 -7 -6 -4-3-2 2 4 5 6 8…

A: Level curves equation y=-6x/5Explanation:

Q: Give two distinct linear functions ƒ and g that satisfy ƒ'(x) = g'(x); that is, the lines have equal…

A: It is given that fx and gx two distinct function.

Q: Find two positive numbers whose product is a such that their sum is minimum.

A: Let the two numbers are x and y So, xy=a y=a/x Their sum= S= x+y We have to minimize S

Q: ind and the slope of the tangent line at the point -1) for the following function. Show all work…

A: Given query is to find the derivative of the expression.

Q: Construct a function that passes through the origin with a constant slope of 1, with removable…

Q: Find the indicated partial derivatives. a3w дz удх a3w д×2ду || = W = X у + 3z

A: w=xy+3z

Q: When would you use a partial derivative? When you only need to differentiate half of a function. O…

Q: Use partial derivatives to obtain the formula for the best least-squares fit to the data points.…

Q: Sketch the level curves of the function f(x, y) = = - y² for c= -4,-1, 0, 1, 4. 9

A: The level curves of a function f(x,y) are formed by plotting points (x,y) in the plane where f(x,y)…

Question

find the partial derivatives. The variables
are restricted to a domain on which the function is defined.

**Problem 8**

Find \( z_x \) if \( z = \frac{1}{2x^2ay} + \frac{3x^{\frac{5}{2}}abc}{y} \).

This problem requires the computation of the partial derivative of \( z \) with respect to \( x \). The expression for \( z \) involves two terms: a rational function and a term with a power of \( x \).

**Problem 5**: Find \(\frac{dz}{dx}\) and \(\frac{dz}{dy}\) if \(z = (x^2 + x - y)^7\).

This problem involves finding the partial derivatives of the function \(z = (x^2 + x - y)^7\) with respect to \(x\) and \(y\).

### Explanation:
- **Partial Derivative \(\frac{dz}{dx}\)**: This involves differentiating the function with respect to \(x\), while keeping \(y\) constant.
- **Partial Derivative \(\frac{dz}{dy}\)**: This involves differentiating the function with respect to \(y\), while keeping \(x\) constant.

Understanding how to find these derivatives is crucial for applications in multivariable calculus, particularly when dealing with functions that depend on more than one variable.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

3.Find the coordinates of the points on the curves y =x3 -3x2-12x+10 with the gradients -2.
Explain Partial Derivatives of a Function of Two Variables?
Consider the linearizable functions as follows. Find new transformed variables (x, y, B or B as you need) to linearize the functions below. 21 y= Biz1+ Bo.
Sketch the level curves of the function f(x, y) = = -8 -7 -6 -5 -4 -3 9 y² for c = -4, -1, 0, 1, 4. 5 6